Peano axioms

In

.

The

provided by Peano axioms is commonly called Peano arithmetic.

The importance of formalizing

).

The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".^[6] The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final, axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema. The term Peano arithmetic is sometimes used for specifically naming this restricted system.

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When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gottlob Frege, published in 1879.^[7] Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.^[8]

The Peano axioms define the arithmetical properties of

N or ${\displaystyle \mathbb {N} .}$ The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S.

The first axiom states that the constant 0 is a natural number:

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number,^[9] while the axioms in Formulario mathematico include zero.^[10]

The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.^[8]

The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S.

Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0.

The intuitive notion that each natural number can be obtained by applying successor sufficiently many times to zero requires an additional axiom, which is sometimes called the

.

The induction axiom is sometimes stated in the following form:

In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Peano arithmetic as first-order theory below.

If we use the second-order induction axiom, it is possible to define

, and can be shown to be unique using the Peano axioms.

Addition

as:

${\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}$

For example:

${\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}$

To prove commutativity of addition, first prove ${\displaystyle 0+b=b}$ and ${\displaystyle S(a)+b=S(a+b)}$, each by induction on ${\displaystyle b}$. Using both results, then prove ${\displaystyle a+b=b+a}$ by induction on ${\displaystyle b}$. The

]

Multiplication

Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as:

${\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}$

It is easy to see that ${\displaystyle S(0)}$ is the multiplicative right identity:

${\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}$

To show that ${\displaystyle S(0)}$ is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:

• ${\displaystyle S(0)}$ is the left identity of 0: ${\displaystyle S(0)\cdot 0=0}$.
• If ${\displaystyle S(0)}$ is the left identity of ${\displaystyle a}$ (that is ${\displaystyle S(0)\cdot a=a}$), then ${\displaystyle S(0)}$ is also the left identity of ${\displaystyle S(a)}$: ${\displaystyle S(0)\cdot S(a)=S(0)+S(0)\cdot a=S(0)+a=a+S(0)=S(a+0)=S(a)}$, using commutativity of addition.

Therefore, by the induction axiom ${\displaystyle S(0)}$ is the multiplicative left identity of all natural numbers. Moreover, it can be shown

addition:

${\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}$.

Thus, ${\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}$ is a commutative semiring.

The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:

For all a, b ∈ N, a ≤ b if and only if there exists some c ∈ N such that a + c = b.

This relation is stable under addition and multiplication: for ${\displaystyle a,b,c\in \mathbb {N} }$, if a ≤ b, then:

• a + c ≤ b + c, and
• a · c ≤ b · c.

Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring.

The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤":

For any

predicate

φ, if

• φ(0) is true, and
• for every n ∈ N, if φ(k) is true for every k ∈ N such that k ≤ n, then φ(S(n)) is true,
• then for every n ∈ N, φ(n) is true.

This form of the induction axiom, called strong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are

—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no least element.

• Because 0 is the least element of N, it must be that 0 ∉ X.
• For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X.

Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N. Thus X has a least element.

A model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and S: N → N satisfies the axioms above. Dedekind proved in his 1888 book, The Nature and Meaning of Numbers (German: Was sind und was sollen die Zahlen?, i.e., "What are the numbers and what are they good for?") that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models (N_A, 0_A, S_A) and (N_B, 0_B, S_B) of the Peano axioms, there is a unique homomorphism f : N_A → N_B satisfying

${\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}$

and it is a

. (This is not the case with any first-order reformulation of the Peano axioms, below.)

The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF.^[15] The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:

${\displaystyle s(a)=a\cup \{a\}}$

The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:

${\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}$

and so on. The set N together with 0 and the successor function s : N → N satisfies the Peano axioms.

Peano arithmetic is

), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.

Interpretation in category theory

The Peano axioms can also be understood using

pointed unary systems

, US₁(C) as follows:

• The objects of US₁(C) are triples (X, 0_X, S_X) where X is an object of C, and 0_X : 1_C → X and S_X : X → X are C-morphisms.
• A morphism φ : (X, 0_X, S_X) → (Y, 0_Y, S_Y) is a C-morphism φ : X → Y with φ 0_X = 0_Y and φ S_X = S_Y φ.

Then C is said to satisfy the Dedekind–Peano axioms if US₁(C) has an initial object; this initial object is known as a

natural number object

in C. If (N, 0, S) is this initial object, and (X, 0_X, S_X) is any other object, then the unique map u : (N, 0, S) → (X, 0_X, S_X) is such that

${\displaystyle {\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}$

This is precisely the recursive definition of 0_X and S_X.

Consistency

When the Peano axioms were first proposed,

Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using

, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.

The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be total. Curiously, there are self-verifying theories that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true ${\displaystyle \Pi _{1}}$ theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a Hilbert-style proof of "0=1").^[23]

All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.^[24] The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.^[25] The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).

First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.

The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of Robinson arithmetic, is sufficient for this purpose:^[26]

• ${\displaystyle \forall x\ (0\neq S(x))}$
• ${\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}$
• ${\displaystyle \forall x\ (x+0=x)}$
• ${\displaystyle \forall x,y\ (x+S(y)=S(x+y))}$
• ${\displaystyle \forall x\ (x\cdot 0=0)}$
• ${\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}$

In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a

. For each formula φ(x, y₁, ..., y_k) in the language of Peano arithmetic, the first-order induction axiom for φ is the sentence

${\displaystyle \forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}}$

where ${\displaystyle {\bar {y}}}$ is an abbreviation for y₁,...,y_k. The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formula φ.

The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative^[27] uses an order relation symbol instead of the successor operation and the language of discretely ordered semirings (axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):

1. ${\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}$, i.e., addition is associative.
2. ${\displaystyle \forall x,y\ (x+y=y+x)}$, i.e., addition is commutative.
3. ${\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}$, i.e., multiplication is associative.
4. ${\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}$, i.e., multiplication is commutative.
5. ${\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}$, i.e., multiplication distributes over addition.
6. ${\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}$, i.e., zero is an identity for addition, and an absorbing element for multiplication (actually superfluous^{[note 3]}).
7. ${\displaystyle \forall x\ (x\cdot 1=x)}$, i.e., one is an identity for multiplication.
8. ${\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}$, i.e., the '<' operator is transitive.
9. ${\displaystyle \forall x\ (\neg (x<x))}$, i.e., the '<' operator is irreflexive.
10. ${\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}$, i.e., the ordering satisfies
    trichotomy
    .
11. ${\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}$, i.e. the ordering is preserved under addition of the same element.
12. ${\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}$, i.e. the ordering is preserved under multiplication by the same positive element.
13. ${\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}$, i.e. given any two distinct elements, the larger is the smaller plus another element.
14. ${\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}$, i.e. zero and one are distinct and there is no element between them. In other words, 0 is covered by 1, which suggests that these numbers are discrete.
15. ${\displaystyle \forall x\ (x\geq 0)}$, i.e. zero is the minimum element.

The theory defined by these axioms is known as PA⁻. It is also incomplete and one of its important properties is that any structure ${\displaystyle M}$ satisfying this theory has an initial segment (ordered by ${\displaystyle \leq }$) isomorphic to ${\displaystyle \mathbb {N} }$. Elements in that segment are called standard elements, while other elements are called nonstandard elements.

Finally, Peano arithmetic PA is obtained by adding the first-order induction schema.

According to Gödel's incompleteness theorems, the theory of PA (if consistent) is incomplete. Consequently, there are sentences of first-order logic (FOL) that are true in the standard model of PA but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as Robinson arithmetic.

Closely related to the above incompleteness result (via

.

Nonstandard models

Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.^[28] The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism.^[29] This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.

When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.

It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as

of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is ω + ζ·η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.

Overspill

A cut in a nonstandard model M is a nonempty subset C of M so that C is downward closed (x < y and y ∈ C ⇒ x ∈ C) and C is closed under successor. A proper cut is a cut that is a proper subset of M. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.

• Foundations of mathematics
• Frege's theorem
• Goodstein's theorem
• Neo-logicism
• Non-standard model of arithmetic
• Paris–Harrington theorem
• Presburger arithmetic
• Skolem arithmetic
• Robinson arithmetic
• Second-order arithmetic
• Typographical Number Theory

Further reading

• Buss, Samuel R. (1998). "Chapter II: First-Order Proof Theory of Arithmetic". In Buss, Samuel R. (ed.). Handbook of Proof Theory. New York: Elsevier Science.
  ISBN 978-0-444-89840-1
  .

• ISBN 978-1-4822-3772-6
  .

• ISBN 978-0-486-49705-1
  .

• ISBN 978-0-486-49073-1.{{cite book}}: CS1 maint: location missing publisher (link
  )

External links

• Murzi, Mauro. "Henri Poincaré". Internet Encyclopedia of Philosophy. Includes a discussion of Poincaré's critique of the Peano's axioms.
• Podnieks, Karlis (2015-01-25). "3. First Order Arithmetic". What is Mathematics: Gödel's Theorem and Around. pp. 93–121.
• "Peano axioms", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Peano's Axioms". MathWorld.
• Burris, Stanley N. (2001). "What are numbers, and what is their meaning?: Dedekind". Commentary on Dedekind's work.

This article incorporates material from PA on

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